91Ó°ÊÓ

Use Gaussian elimination to find the complete solution to each system of equations, or show that none exists. $$\left\\{\begin{aligned} w-3 x+y-4 z &=4 \\ -2 w+x+2 y &=-2 \\ 3 w-2 x+y-6 z &=2 \\ -w+3 x+2 y-z &=-6 \end{aligned}\right.$$

Perform each matrix row operation and write the new matrix. $$\left[\begin{array}{rrr|r} 2 & -6 & 4 & 10 \\ 1 & 5 & -5 & 0 \\ 3 & 0 & 4 & 7 \end{array}\right] \quad \frac{1}{2} R_{1}$$

Use the fact that if \(A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right],\) then \(A^{-1}=\frac{1}{a d-b c}\left[\begin{array}{rr}d & -b \\ -c & a\end{array}\right]\) to find the inverse of each matrix, if possible. Check that \(A A^{-1}=I_{2}\) and \(A^{-1} A=I_{2}\) $$A=\left[\begin{array}{rr} 2 & 3 \\ -1 & 2 \end{array}\right]$$

Perform each matrix row operation and write the new matrix. $$\left[\begin{array}{rrr|r} 3 & -12 & 6 & 9 \\ 1 & -4 & 4 & 0 \\ 2 & 0 & 7 & 4 \end{array}\right] \quad \frac{1}{3} R_{1}$$

Use the fact that if \(A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right],\) then \(A^{-1}=\frac{1}{a d-b c}\left[\begin{array}{rr}d & -b \\ -c & a\end{array}\right]\) to find the inverse of each matrix, if possible. Check that \(A A^{-1}=I_{2}\) and \(A^{-1} A=I_{2}\) $$A=\left[\begin{array}{rr} 0 & 3 \\ 4 & -2 \end{array}\right]$$

Find the following matrices \(\begin{array}{lll}\text {a. } A+B & \text { b. } A-B\end{array}\) \(c,-4 A\) \(d .3 A+2 B\) $$A=\left[\begin{array}{lll}6 & 2 & -3\end{array}\right], B=\left[\begin{array}{lll}4 & -2 & 3\end{array}\right]$$

Use Gaussian elimination to find the complete solution to each system of equations, or show that none exists. $$\left\\{\begin{array}{rr} 3 w+2 x-y+2 z= & -12 \\ 4 w-x+y+2 z= & 1 \\ w+x+y+z= & -2 \\ -2 w+3 x+2 y-3 z= & 10 \end{array}\right.$$

Use Cramer's Rule to solve each system. $$\left\\{\begin{array}{rr} x-2 y= & 5 \\ 5 x-y= & -2 \end{array}\right.$$

Perform each matrix row operation and write the new matrix. $$\left[\begin{array}{rrr|r} 1 & -3 & 2 & 0 \\ 3 & 1 & -1 & 7 \\ 2 & -2 & 1 & 3 \end{array}\right] \quad-3 R_{1}+R_{2}$$

Use Gaussian elimination to find the complete solution to each system of equations, or show that none exists. $$\left\\{\begin{array}{l} 2 x+y-z=2 \\ 3 x+3 y-2 z=3 \end{array}\right.$$